Our proof is a simple consequence of an extension of a protocol from
[1]} for embezzlement of entanglement, which can be described
as follows.
Consider a scenario in which the direct transformation of a state
\(\rho \in {\cal D}(d)\)  into another state \(\sigma \in {\cal D}(d)\)  is
impossible. For example, the transformation could be forbidden because
\(\rho \)  and \(\sigma \)  contain different amounts of a quantity that is
conserved under the transformation.
For example, the 0- and 1-photon number states are not interconvertible
if energy is conserved.
Alternatively, the transformation can be impossible because the
allowed set of operations could be limited.
For example, if the transformations are limited to the set of local unitaries,
and
\(\rho \)  and \(\sigma \)  contain different
amounts of entanglement entropy, which is known to be conserved under local unitaries, then \(\rho \)  and \(\sigma \)  cannot be transformed into one another exactly.
The transformation, although impossible to realize exactly, is known to be realizable approximately via a protocol from reference [1]}, which uses a
“catalyst” state \(\Gamma \in {\cal D}(d^{n})\) , where
\(n\)  determines the accuracy of the transformation:
\(\Gamma = {1 \over n-1} \sum _{r=1}^{n-1} \rho ^{\otimes r} \otimes \sigma ^{\otimes n-r}.\) 
